Question

In Problems \(57-64\), use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places. $$ \left\\{\begin{array}{l} y=x^{2 / 3} \\ y=e^{-x} \end{array}\right. $$

Short answer

The solutions are approximately (0.91, 0.40) and (-0.69, 2.00).

Step-by-step solution

- Understand the equations

The system of equations consists of two functions: 1. The first equation is \( y = x^{2/3} \).2. The second equation is \( y = e^{-x} \). Both functions need to be graphed to find their points of intersection.

- Graph the first equation

The first equation is \( y = x^{2/3} \). Use the graphing utility to plot this function. It has a typical root curve shape.

- Graph the second equation

The second equation is \( y = e^{-x} \). Use the graphing utility to plot this function. It is a decaying exponential curve that decreases from left to right.

- Find intersections

Identify the points where the two graphs intersect. These intersection points represent the solutions to the system of equations. The x-coordinates of these intersection points need to be rounded to two decimal places.

- Verify the solutions

Verify the solutions by substituting the x-coordinates found into both equations to ensure that the resulting y-coordinates match.

- Report the solutions

List the rounded coordinates of the intersection points as the solutions to the system of equations.

Key concepts

graphing utility

A graphing utility is a powerful tool used to visualize mathematical functions and their intersections. This can be a graphing calculator or software like Desmos, GeoGebra, or even an online plotter. Using a graphing utility helps you effortlessly draw complex functions and observe their behavior over a range of values.
When using a graphing utility, enter each function into a different slot and ensure that the viewing window is set correctly to capture the intersection points. This visual representation is essential in understanding the dynamics of the equations.

intersection points

Intersection points occur where two different curves meet on a graph. These points are vital as they represent the solutions to a system of equations. In this exercise, the intersection points are where the curves of the functions \( y = x^{2/3} \) and \( y = e^{-x} \) cross each other.
To determine these points, plot both functions on the same graph. Look for places where the lines overlap. These coordinates are the solutions to the system. Each x-coordinate represents an x-value solution, and each y-coordinate must be verified to ensure they satisfy both equations.

exponential function

An exponential function is one where the variable appears in the exponent, such as \( y = e^{-x} \). This function describes rapid changes, either growth or decay, depending on whether the exponent is positive or negative. In this problem, \( y = e^{-x} \) represents an exponential decay function because it decreases as x increases.
When graphing this function, observe that it starts high when x is negative and approaches zero as x becomes positive. This behavior is crucial for identifying where it interacts with other curves and determining the intersection points.

root curve

A root curve is a type of curve that results from an equation involving a fractional exponent. For example, \( y = x^{2/3} \) is a root curve. It resembles a parabola but with a softer bend near the origin. The fractional power signifies the curve behavior, which in this case is a cube root followed by squaring.
When graphing root curves, note how they open up symmetrically around the y-axis if the exponent numerator is even. Understanding the root curve's shape simplifies finding its intersections with other functions, like exponential functions.

decimal approximation

Decimal approximation involves rounding numbers to a specific number of decimal places for simplicity and clarity. In this exercise, after identifying the intersection points where the curves meet, we express the x-coordinates of these points to two decimal places.
This process entails visual inspection and using the graphing utility’s precise tools to find values as close to true as possible, then rounding them appropriately. Decimal approximation helps us present answers in a more digestible and standard form, enhancing readability and comprehension.